Problem 16 · 2025 Math Kangaroo
Medium
Logic & Word Problems
casework
In a room there are 10 more people who always tell the truth than there are people who always lie. Everyone in the room was asked, “Are you telling the truth?” and all 20 people answered yes. How many liars are in the room?
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Answer: B — 5
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Hint 1 of 2
Both truth-tellers and liars answer 'yes' to 'Are you telling the truth?' — so the answers tell you nothing new.
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Hint 2 of 2
Just use that truth-tellers outnumber liars by 10 and there are 20 people.
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Approach: solve the count from the two given totals
- A truth-teller says 'yes' truthfully; a liar also says 'yes' (lying), so all 20 say yes regardless.
- If liars = L, truth-tellers = L + 10, and (L+10) + L = 20.
- So \(2L = 10\), giving \(L = 5\) liars, which is (B).
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